> I know that 'professor' in American English doesn't mean 'the head of a
> university department' which is the meaning elsewhere ...

A professor is anyone with a degree that teaches a class.

> and I know that
> 'college' is used to mean what I would call 'university' ...

A university is divided into several colleges (The College of Engineering,
The College of Math, C of Liberal Arts, C of the Arts, &c).  Colleges are
then further subdivided into Departments (The Department of Mechanical Engineering,
The Dept. of Civil Eng., The Dept. of Electrical Eng., &c).

> and I even
> know - because my American friends have told me - that American
> universities have special names for student year levels instead of just
> 'first year', 'second year', 'third year' as it is here ...

Freshman, Sophomore, Junior, Senior

> ... but this I don't know. What exactly is a dean? We don't have them.

A Dean is the head of a college.